Physics On the Symplectic Structure of General Relativity - Project Euclid

Communications in Commun. Math. Phys. 86, 55-68 (1982)

Mathematical

Physics © Springer-Verlag 1982

On the Symplectic Structure of General Relativity Abhay Ashtekar 1 ' 21 and Anne Magnon-Ashtekar3§ 1 Physics Department, Syracuse University, Syracuse, NY 13210, USA* 2 Departement de Physique, Universite de Clermont-Fd., F-63170 Aubiere, France 3 Departement de Mathematiques, Universite de Clermont-Fd., F-63170 Aubiere, France

Abstract. The relation between the symplectic structures on the canonical and radiative phase spaces of general relativity is exhibited.

1. Introduction

There are available in the literature, two Hamiltonian descriptions of general relativity. The first and the more established one is based on spacelike hypersurfaces and uses the initial value formulation of general relativity and the Dirac theory of constrained systems [1,2]. Over the years, this formulation has been systematically developed and refined by several authors and has shed considerable light on the structure of Einstein's theory. (See, e.g., [3].) In particular, these investigations have brought out the role of the Arnowitt-Deser-Misner [4] energy-momentum as the generator of space-time translations [5] and have paved the way for canonical quantization of gravity [3]. The second Hamiltonian description became available more recently [6]. It is based on null infinity [7] and uses techniques from the gravitational radiation theory in exact general relativity, (See especially, [8] and [9].) Here the focus is on the radiative aspects of the gravitational field; the phase space is the space of radiative modes. This description has also given one new insight. In particular, fluxes of energy-momentum and angular momentum carried away by gravitational waves have been shown to be the generators of the Bondi-Metzner-Sachs (BMS) group, the asymptotic symmetry group at null infinity [10]. More importantly, the formulation has enabled one to carry out the asymptotic quantization of the non-linear gravitational field [6,11]. In view of this situation, it is natural to ask for the relation between the two descriptions. Apart from its intrinsic interest, such an analysis would clarify several issues which arise in the two frameworks separately. For example, since the radiative f

Alfred P. Sloan Research Fellow. Supported in part by the NSF contract PHY80-08155 and by a grant from the Syracuse University Research and Equipment Fund § Supported in part by credits ministeriels, tranche speciale * Permanent address

0010-3616/82/0086/0055/S02.80

56

A. Ashtekar and A. Magnon-Ashtekar

phase space is not constructed from a cotangent bundle over a configuration space, the symplectic tensor field thereon had to be simply postulated [10]. Here, one was guided by general considerations such as the requirement that the Poisson bracket between the basic variables should have the dimensions of action, that one should obtain the correct results in the weak field limit, and that the expression of the symplectic structure should fit in the pattern suggested by the spin zero and one fields. However, one could not show that these considerations suffice to determine the symplectic tensor field uniquely. It is therefore desirable to have as strong an evidence as possible supporting the choice that was made. A strong— perhaps the strongest possible—evidence would be that the chosen symplectic structure is, in an appropriate sense, the same as the one on the canonical phase space. The canonical approach would also be enriched from the analysis of its relation to the radiative framework. For example, the canonical quantization programme has met with severe difficulties in the construction of a Hubert space of states (or a substitute thereof). In the approach based on the radiative phase space, on the other hand, these difficulties do not arise: one can readily construct not only the Fock spaces of asymptotic gravitons but also the Hubert spaces required to handle the infrared problems, i.e., which are analogous to the charged sectors in quantum electrodynamics [12]. Therefore, an understanding of the relation between the two phase spaces may give one considerable insight in the Hubert space problem of canonical quantization. In particular, the analysis may shed light on the nature of the (canonical) quantum vacuum, which, one now suspects, may not be simply a gaussian peaked at the flat metric. The purpose of this paper is to provide the first steps towards establishing the relation between the two phase spaces. At an intuitive level, one may divide the problem into two parts: differential geometric issues and functional analytic difficulties. In a broad sense, this paper resolves the first part. More precisely, we shall assume that globally hyperbolic, vacuum, asymptotically flat, horizon-free space-times exist and show that each such space-time leads to a natural symplectic structure preserving identification of a point of the canonical phase space with a point of the radiative phase space. The main obstacle in relating the two phase spaces is, of course, that whereas the canonical phase space is constructed from initial data sets on space-like surfaces, the radiative phase space consists of certain equivalence classes of connections on null infinity, J} Therefore, to exhibit the relation between the two, we shall introduce a structure which can interpolate between the two regimes: the symplectic vector space of linearized gravitational fields on a globally hyperbolic asymptotically flat, vacuum space-time without horizons. This introduction serves the following purpose. A linearized solution induces on any Cauchy surface a set of linearized Cauchy data and may be therefore regarded as a tangent vector at a point of the canonical phase space. As one might expect, this identification preserves the symplectic structure: We shall show explicitly that the symplectic structure on the space of linearized solutions (off a fixed background) reduces to the symplectic structure evaluated at the tangent space of any point of the canonical phase space 1

Throughout this paper, the symbol J will stand for future or past null infinity

Symplectic Structure

57

corresponding to the given background, when the linearized fields are identified 2 with their initial data. On the other hand, if the background is asymptotically flat at null infinity, each linearized solution also defines a linearized connection on ,/, and therefore, a tangent vector to the phase space of radiative modes at the point corresponding to the given background. We show that this identification is also symplectic structure preserving, thereby exhibiting the equality between the canonical and the radiative symplectic structures. To summarize, the difficulty in relating the two frameworks is overcome by first recognizing that a linearized solution to Einstein's equation defines a tangent vector at suitable points of both phase spaces and that the symplectic structure, being a tensor field, is completely determined by its action on the tangent vectors, and then letting the linearized solutions do the desired interpolation between the space-like surface and J. 2. Preliminaries This section is divided into four parts. The first summarizes the usual Hamiltonian formulation of general relativity; the second outlines the structure available on the space of radiative modes in exact general relativity; the third describes the phase space of linearized gravitational fields on a vacuum, globally hyperbolic background space-time and the fourth recalls certain results on the asymptotic behavior of these linearized solutions. 2.1 The Hamiltonian Formulation of General Relativity Fix a C 00 3-manifold Σ and consider thereon pairs {qab,pabmnr) consisting of C°° positive-definite metrics qab and Crχ) tensor fields pabmm. such that pabmnr = P{ab\mnrV3 We shall assume that either Σ is compact or the pairs (q, p) are asymptotically flat in a suitable sense. The space Γ of these pairs has the structure of a cotangent bundle. It therefore possesses a natural symplectic tensor field Ω: Ω\{qJ&

β); (α', β')) = j (αβbj8'β*mBΓ - α''abβabmnr)dS™\

(1)

where (α,β) represents a tangent vector to Γ at (q,p). (Thus, aab is a symmetric ab ab tensor field and β mnr has the symmetries of p mnr). Denote by Γ the "constraint submanifold" of Γ consisting of pairs (q,p) satisfying the following equations: ab

DaP

mnr

= 0;

and,

(2.a) h mP

™ = 0,

(2.b)

where D and 0ί are, respectively, the derivative operator and the scalar curvature of qab, and where indices are raised and lowered by qab. Each point of Γ represents a permissible data for Einstein's vacuum equation: qab is the intrinsic metric on Σ and nab\=(l/6){pabmnr-^qabpccmnr)εmn\ the extrinsic curvature, where εabc is the abc unique 3-form on Σ defined by ε zabc = 3!. Denote by Ω the pull-back to Γ of Ω. 2 3

Because of gauge problems, this result is not as straightforward as one might have expected One often uses tensor densities of weight one in place of tensor fields pabmnr

58


Thus, we have: Ω\{qs{{ά,β)\(a'Jf)) = Ω\^-p)((d,β);(df,β')) for allvectors (άj) and (α',/0 tangential to Γ at (q,p). This Ω is, however, degenerate: Ω((d,β);(ά',β')) = 0 for all tangent vectors (α',/0 to Γ if and only if (α,β) is the (restriction to the point {q, p) of the) Hamiltonian vector field on Γ generated by the constraint function CNNa{q,PΪ = j dSîN{\pahuυwparw-hv\uυ^Γ

~ #)**„ " 2NbDapabmnrl

(3)

Here, the lapse N and the shift Na are C 00 fields on Σ, which, in the asymptotically 4 flat case, vanish at a suitable rate at infinity. The reduced phase space of general relativity is the "manifold of orbits" of these constraint vector fields, restricted to Γ. Denote it by f. The tangent space T at any point (q, p) of Γ can be identified with the quotient T/S of the tangent space T at any point (q, p) of Γ (which projects down to (q, p) in Γ) by its subspace S which is spanned by the constraint vector fields. Hence, f inherits from Γ a weakly non-degenerate symplectic structure Ω: β | ( ^ ( M ) ; ( α ' J ' ) ) = Ω\{^}(&β);(ά\βn

(4)

where (ά,β) is any element of T which projects to (ά,j?) in T. (For details, see, e.g. [13], [14], or [15].) 2.2. The Phase Space of Radiative Modes in the Exact Theory Fix a 3-manifold J, topologically S2 x JR, equipped with a collection of pairs of C °° fields (qab, na\ with qab symmetric, satisfying the following conditions: i) qab Vb = 0 if and only if Va is proportional to na; ii) £?nqah = 0; iii) (q,n) and (q,ή) are both in the collection if and only if there exists a function ω on J such that qab = ω2qab,na = ω~γna and S£nω = 0; and, iv) na is a complete vector field and the space of its orbits is diffeomorphic to S2. Thus «/ is equipped with the "universal structure" of Penrose's null infinity [16]. Fix a conformal frame—i.e. a pair (q,ή) from the collection—on J and denote by V the affine space of torsion-free connections D o n / satisfying Daqbc = 0

and Danb = 0.

(5.a)

Finally, introduce the following equivalence relation on # : D~D'

if and only if

(Da - D'a)Kb = fqabncKc

(5.b)

for any function / o n / (independent of the choice of Ka). Denote the space of equivalence classes {D} by Γ. This is the required space of radiative modes of the non-linear gravitational field in exact general relativity. Let us examine the structure available on Γ. It is easy to show that connections D and D both belong to # if b and only if there exists a symmetric tensor field Σah with Σahn = 0 such that c (Da - DJKb = ΣabKcn for all Ka on «/. Hence it follows that the difference between any two elements {D} and {D} of Γ can be completely characterized by the trace-free part, yab, of Σab. Thus, Γ has the structure of an affine space; by fixing any one 4 N and Na have to fall-off "at least as 1/r" for CNtNa to be a C 1 function on Γ, i.e., to generate a Hamiltonian vector field. According to the Dirac theory of constrained systems, such vector fields generate gauge motions


59

point {D0} as the "origin," one can coordinatize Γ by tensor fields yab satisfying Ίab = Ί(ab)> y°bnb = 0, a n d Ίab^^^ ( τ h e t w 0 independent components of yab represent the two radiative modes of the gravitational field.) Finally, the following symplectic tensor field has been introduced on Γ: β|{D}(y9 / ) : = ί (yab^nyfcd

- yfab^nycdkac^mnrdsmnr.

(β)

Here, qab is any "inverse" of qab and εabc is the unique 3-form on J satisfying abc = 3! where εabc is defined by £abcεmnpqamqbn = ncnp [17]. It is easy to verify 8abcs that Ω is conformally invariant and weakly non-degenerate and has the dimensions of action. These properties, together with the pattern suggested by the symplectic structures of zero rest mass, spin zero and one fields, provided the original motivation behind this choice of Ω. Further evidence came from the fact that the action of the BMS group of J induces motions on Γ which preserve Ω and the Hamiltonians generating these canonical transformations on (Γ,Ω) provide the formulae for fluxes of energy-momentum, supermomentum and angular momentum carried away by the gravitational waves. (For details, see [10] and [11].) 2.3. The Symplectic Tensor of Linearized Gravitational Fields Fix a globally hyperbolic, vacuum space-time (M,gab). Let us suppose that this space-time admits a foliation by Cauchy surfaces which are either compact or asymptotically flat at spatial infinity in a suitable sense. Denote by hab a solution to the linearized vacuum equation: VmVmhab + 2Ramhnhmn

- 2V(flV H ( V - \hδh)m) = 0,

(7)

d

where V and Rabc are respectively the derivative operator and the Riemann tensor on (M, gab). It is easy to verify that hab = V{aζb) satisfies Eq. (7) for arbitrary vector fields ζb. Such solutions represent "pure gauge" linearized fields. Denote by V the space of C 00 solutions to Eq. (7) the intersection of whose support with any Cauchy surface is compact and by V the quotient of V by the subspace containing pure a gauge fields V(αζb), where the support of ζ has a compact intersection with any Cauchy surface. Consider the skew tensor ώ on V, defined by: mn

ώ(fc, ft') = 3 J ε \(hmsVnh'pr

- h'^JiJdS*".

(8)

Σ

It is easy to check, using Eq. (7), that the integrand is a curl-free 3-form whence the integral is independent of the choice of the Cauchy surface Σ. Using this f property (or, by direct substitution in Eq. (8)) one can show [18] that ώ(h,h ) = 0 for all Wab in V if hab is a pure gauge field, i.e., if hab = V{aζb) for some ζb. Hence ώ induces a skew tensor ώ on V: ώ({ft},{ft'}) = ώ(ft,ft/),

(9)

where {h} in V denotes the equivalence class of elements of V to which h in V 5 belongs. This ώ can be shown to be weakly non-degenerate : ώ({h}, {h'}) = 0 for 5

This holds provided the background metric gab does not admit Killing fields near spatial infinity

60

A. Ashtekar and A. Magnon-Ashtekar 1

all {h} in V if and only if {h } = 0. Thus, (V,ώ) is a symplectic vector space. This structure has been exploited in the construction of conserved quantities from linearized fields in the case when (M,gab) admits a Killing field. (For details, see [18] and [19].) 2.4. Asymptotic Behavior of Linearized Fields The connection between the symplectic structure Ω of the canonical phase space and Ω of the radiative modes will be established in the next section using the symplectic vector space (F,ώ) of linearized fields. Therefore, we shall need information about the asymptotic behavior of the linearized fields hab at null infinity. Fortunately, an extensive analysis of this issue already exists in the literature. We shall therefore merely quote the required result: Theorem [20]. Let.(M,gab) be an asymptotically flat and empty space-time in the sense of Geroch and Horowitz [27]. Denote by (M = M u / , ^ = u2gab) o n e °f its conformal completions in which J* is divergence-free (i.e., in which α satisfies VmVmα = 0 on J>). Let h'ab be a C°° solution to the linearized vacuum equation in a neighborhood of J>, the intersection of whose support with some Cauchy surface is compact. Then, there exists a solution hab, related to h'ab by a gauge transformation, such that hab: = and gab^nhab vanishes on J, where, na = faoc is the null normal to J. 3. Relation between Ω and Ω This section is divided into two parts. In the first, we investigate the relation between the reduced phase-space (f,Ω) and the symplectic vector space (V,ώ) constructed from the linearized solutions, and, in the second, that between (V,ώ) and the phase-space (Γ, Ω) of radiative modes. 3.1. Relation Between (f, Ω) and (V, ώ) Consider a globally hyperbolic, vacuum space-time (M,gab). Fix a Cauchy surface Σ and denote by (qab, pabmnr) the point of the constraint submanifold of the canonical space Γ defined by the initial data of gab on Σ. We shall assume that either Σ is compact or that the pair (qab, pabmnr) is asymptotically flat in a suitable sense and show that there exists a natural one to one mapping Ψ from V to the tangent space f{qp) of f at (qab, pabmnr) and that Ψ maps Ω to ώ. Lemma 1.1. Given a linearized solution h'ab in V, there exists a gauge related solution hab in V satisfying habnb = 0, where nb is the unit normal to Σ and = denotes equality restricted to Σ. Proof. Consider a scalar field N and a vector field Na on M satisfying a

a

a

b

N = 0, N = 0,2n WaN = n n h' a

a

and

ab

a

bm

a

q n VaNb a

a bm

= - nq h

'ab.

Set hab = Wab + 2¥(aζb) with ξ = Nn + N , where n now denotes an unit time-like, hypersurface orthogonal extension to M of the normal to Σ. This hab is again in V and satisfies habnb = 0. Π


61

Denote by V the subspace of V consisting of linearized solutions hab satisfying b habn = 0. By Lemma 1.1, the natural mapping, hab->{hab] from V to V is onto. (However, it is not one to one: given hab in V,hab + 2V{aξb) is again in V provided a a b ξ satisfies n V(aξb) = 0.) The gauge condition habn = 0 is introduced for convenience only; it simplifies the task of computing the first order changes in the intrinsic ab metric qab and the extrinsic curvature π of Σ. A simple calculation yields the ab

a

following result: δqab = hab and δπ to πab via

b

mc nd

ab

= \q cq d^n(q q hmn).

_ ίΉab _ nab P mm — \Jl

Since p

.

mm

is related

ab\F )bmnr>

πa

llί

i

one has, b

±

l

b

d

U

Û

τrnabY\p

, -foab . ft/.-αb _

\nab

Thus, the natural mapping Ψ from V to the tangent space T ( ^ p ) of the constraint sub-manifold Γ at the point (qab, pabmnr) is given by Ψ(hJ = (oiab9βabnmrl flb

(Π)

flb mπκ

where aab = ^^flb = feflb and β mm. = (5p has the expression given in Eq. (10). We now ask if Ψ gives rise, naturally, to a mapping Ψ from V to t{ίιf)Y i.e., if Ψ maps "pure gauge" linearized fields hab = 2V{aξb) to a tangent vector in Γ representing an infinitesimal "gauge motion." An affirmative answer is given by the following result: Lemma 1.2. Let hab = 2V{aξb) be in V, where the intersection of the support of ξb with any Cauchy surface is compact. Then, Ψ(hab) = (aab, βabmnr) is the restriction to (Qab>Pabmnr) °f t n e Hamiltonian vector field generated on Γ by the constraint function CNtNa(q,p) (ofEq. (3)), where the lapse N and the shift Na are given by N = — ξana and Na = qamξm. (Here, as before, na is the unit (future-directed) normal to Σ and qam = δam -f nanm is the projection operator associated with Σ). Proof Since hab belongs to V, naV{aξb) = 0. This implies that VaN = 0 and ^nNa Using these properties, it is straightforward to show that ψ(L

\ — (0 /VTΓ

-4- (P n

N( — &ab

-I- Φπab

-I- ~nab

n1

= 0.

cd

$

_lna btcdf\p i a> nab \ 3F tcdfP )£mnr + ^ NP mmh

where nab is the extrinsic curvature of Σ and &ab is the Ricci tensor of qab. On the other hand, using the fact that N and Na have compact spatial support, one can show [14] that the restriction to (qab,pabmnr) in Γ of the Hamiltonian vector field generated by CN Na(q,p) is given by:

(εmJδC/δfbmM-δC/δqab)εmnr)

= (j(Pah^

- \qabp

V

ah

N( - M* + \PΛq + y

_ J_ n c ab\Q , cp ab n d uvw 24rP cuvwP d Q iεnmr+îvP mnr

62

A. Ashtekar and A. Magnon-Ashtekar ab

Since (qab,p

)

mnr

satisfy the constraint equation (2.b), we have ab

Ψ(hab) = ((δC/δp

( - δC/δqab)εmnr).

)εmnr,

mnr

D

Remarks, (i) Since hab is an element of V, it follows that V(aξb) has compact spatial a a support. Note, however, that ξ need not share this property: ξ may be a Killing field outside a bounded world tube. This is why an explicit condition on the a support of ξ had to be imposed in the statement of Lemma 1.2. (ii) The a requirements on the support of hab and ξ can be weakened substantially without altering the essence of the results contained in Lemmas 1.1 and 1.2. Let us suppose, for example, that (M,gab) is asymptotically flat at spatial infinity in the sense of [22] and denote by (M = Mvi°,gab = oc2gab\ one of its conformal completions. Consider on (M,gab) linearized solutions hab which preserve the requirements of asymptotic flatness to first order. (Thus, hab need not have compact spatial support, it may fall off only "as 1/r" at space-like infinity.) Denote this space by V'. Consider as gauge those elements of V' which are of the form V{aξb) where, on M, ξa falls-off as α 3 / 2 . (Thus the one parameter group of diffeomorphisms generated by ξa is asymptotically identity.) Denote by V' the quotient of V' by its gauge subspace. We could have used V' and V' in place of V and V in Lemmas 1.1 and 1.2 and the above enlarged class of vector fields ξa in Lemma 1.2. Note, incidentally, that if ξa generates (non-identity) spin symmetries [22], hab = V{aξb) belongs to V' but not to its gauge subspace. Hence Ψ(hab) is not the restriction to (qab,pabmnr) of the Hamiltonian vector field generated by CNiNa for any choice of N and Na: If ξa falls off slower than α 3 / 2 , CN>Na defined by'iV = ξana,Na = q\ξb fails to be C 1 on Γ and therefore cannot lead to a Hamiltonian vector field. Lemmas 1.1 and 1.2 imply that Ψ given by Ψ°{h} = {Ψ°h}{eT/S) is a welldefined mapping from V to the tangent space T^^ of the reduced space f. We now ask if this mapping preserves the symplectic structure. One has: Theorem 1. ώ({h}9{h'}) = 0(φo{h}9ψo{h'})

for all {h} and {W} in V.

Proof. By Lemma 1.1, we can choose from the equivalence classes {h} and {h'} elements hab and h'ab in V. Let us make such a choice. Then, m

ώ(h, h')=3\

-

s "oq(hmsVnh'pr

h'msVnhpr)dS«"

Σ ~2)b

q\nmsynn

pr

n

msynnpr)b

n U

t

V

Σ

Σ

= - i j {hJ?n(h'abq"b) - h"»J?nh'ab + hh'abπ"» -

(h~h')}dVΣ,

Σ

where in the last step, we have used the gauge condition habnb = 0 and h'abnb = 0. On the other hand, one has

Ω(Ψ(h), Ψ(h')) Ξ J (aabβ'abmnr

*'lΛpΛnm)dSr'

-

Σ

Γ-i- ^ Σ

h'

amΠbn

Π

±nab q? (h'cdn

\ -4-

τrh'ab


63 — _ 1 Γ [1% Cf (h'cdn \ — 2 J 1/*°2- n\ι Hcd) Σ

hab (P h' -X-hW rrab ^ °^ n^ ab ' ^ ^ ab

Thus 6 , ώ(h,h') = Ω{Ψ(h\ ψ{h')\ Hence, by Eqs. (4) and (9), we have ώ({h}, {hf}) = /j}7 Ψ°{ii'}) for all {h} and {A'} in K • Remarks, (i) Consider the case when (M, gab) admits no Killing field in a "neighborhood of spatial infinity." Then, the symplectic structure ώ on V is weakly non-degenerate. From Theorem 1 it now follows that the mapping Ψ must be one to one. (ii) If we had enlarged V to V' as indicated in the remark following Lemma 1.2, the mapping ψ would have been an isomorphism between V' and Γ. Without this enlargement, however, Ψ is only an imbedding of V into Γ. 3.2. Relation Between (V, ώ) and (Γ, Ω) Let us now make further assumptions on the background space-time {M,gab)\ let us suppose that ( M , ^ ) is asymptotically flat at null infinity and is free of horizons, i.e., has the property that the causal past of the future null infinity is all of M. We shall now show that the natural mapping from V to the tangent space T{D} of Γ (at the point {D} corresponding to the physical metric gab) sends Ω to ώ. Fix a conformal completion (M = MvJ,gab = a2gab) of {M9gab) in which J is α divergence-free, i.e., in which α satisfies V Vαα = 0 (nabnb>

a n d

ot~1habnanb

have C 0 0 limits on J and $£nhabqab

= 0, where na and

qab are respectively the null normal and the degenerate intrinsic metric on J and qab is any "inverse" of qab. We wish to compute the linearized connection "{D}" induced on J by hab. Let us first recall how one obtains the equivalence class {D} on J starting from the space-time metric gab. Given a covector field K6 on J, set DaKb = VαXb, 00 where Kb is any C extension of Kb to a neighbourhood of J in 1VΪ, v ' i s the derivative operator compatible with gab on M and where the arrow stands for 00 "pull-back to JΓ Since any two C extensions, Kb and K'b of Kb are related by 00 K'b - Kb =fVba + ocVb, where / and Vb admit C limit to J, and since α = 0 and VflVbα = 0, it follows that DaKb is independent of the choice of the extension of Kb. 1 By requiring that DJ be the gradient of / for all C functions / o n / and that D be linear and satisfy the Leibnitz rule with respect to outer product, one can extend the action of Da uniquely to arbitrary tensor fields within J. Thus, D may be thought of as the pull-back to J of V. It automatically satisfies Eq. (5.a). Under the permissible conformal rescalings oϊgab9 both V and D change. One is therefore led to introduce the equivalence relation of Eq. (5.b): Whereas D refers to a specific conformal completion (M,gab) oi(M,gab\ {D} refers to all permissible completions. More specifically, {£>} represents the radiative modes of the gravitational field associated with the physical metric gab. [11] 6 In the light of this result, Lemma 1.2 may seem superfluous. However, it is not: we have not shown that Ψ is on to

64


Let us now consider a one-parameter family of vacuum metrics gab(λ) such that 9ab(ty = ΰab> Λe given background, and —gab(λ)\λ

=0

= hab, the given linearized

λ

7

metric. Denote by V the one parameter family of connections associated with = a n λ Qa4J) QPQJΆ* d by D the corresponding family of connections on J. It is easy to check that the first order changes in the connections V and D induced by hab are given by: VbΩ2had

-

and (δD)aKb: = —λDaKb\{λ=0)

= Um(-2a)hab(gcdVda)Kc

= - 2habncKc,

where Kb and Kb are, respectively, arbitrary covector fields on M and J, na is the null normal to J and hab = Lim ahab. It now follows that the tangent vector to Γ at the point {D}, defined by the linearized perturbation hab is given by 8 the trace-free part oϊhab: (12)

δ{D}^yab = hab-^mnq^qab.

Thus, we have a natural mapping Φ from the subspace V of F, consisting of linearized solutions satisfying the Geroch-Xanthopoulos gauge, to the tangent space T{D} of Γ at the point {D}\ (13)